The Impedances, Angular Velocities and Frequencies of Oscillating-Current Circuits

Corresponding to the usual angular velocity (2 Π times the frequency) of an alternating current is the generalized angular velocity of an oscillating current. The generalized velocity is a complex quantity; the real portion determining the damping constant, the imaginary portion the frequency of the current. The author shows that the oscillation impedances of resistances, inductances and capacities are formed in the same way from generalized angular velocities as from the usual angular velocity. The oscillation impedance of any circuit or system of circuits is found by the usual law of resistances for continuous currents, due regard being paid to the rules of complex quantities. It is then shown that free oscillations of any system of circuits select such angular velocities as to reduce the total oscillation impedance to zero. A number of cases of parallel and series oscillating circuits are treated by this method with much simplicity. The total oscillation admittance at a knot point is shown to be zero, as also is the sum of the instantaneous oscillation-impedance drops around a closed loop. The instantaneous discharge power in any oscillation impedance is readily derived and shown to be zero in a pure oscillation system. The problem of coupled circuits is given a preliminary treatment by these methods.